problem:Suppose that x and y are vectors and M is a subspace in a vector space V;
let K be the subspace spanned by M and x, and let K be the subspace spanned by M and y.
Prove that if y is in K but not in M, then x is in K.

attempt:So, K is the set of all linear combinations of elements in M and the vector y.
K is also the set of all linear combination of elements in M and the vector x.
This definition of K makes me think that x is always in K.
What am I missing here?

Thanks!

Feb 5th 2010, 12:36 AM

flyingsquirrel

Hi,

Quote:

Originally Posted by Mollier

problem:Suppose that x and y are vectors and M is a subspace in a vector space V;
let K be the subspace spanned by M and x, and let K be the subspace spanned by M and y.
Prove that if y is in K but not in M, then x is in K.

It does not make sense : $\displaystyle K$ is at the same time the subspace spanned by $\displaystyle M$ and $\displaystyle x$ and the subspace spanned by $\displaystyle M$ and $\displaystyle y$... but these subspaces may be different !

Here is what I think they meant :

Quote:

problem:Suppose that $\displaystyle x$ and $\displaystyle y$ are vectors and $\displaystyle M$ is a subspace in a vector space $\displaystyle V$;
let $\displaystyle K_x$ be the subspace spanned by $\displaystyle M$ and $\displaystyle x$, and let $\displaystyle K_y$ be the subspace spanned by $\displaystyle M$ and $\displaystyle y$.
Prove that if $\displaystyle y$ is in $\displaystyle K_x$ but not in $\displaystyle M$, then $\displaystyle x$ is in $\displaystyle K_y$.